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Since image analysis techniques have come to maturity, mesh analysis has remained challenging requiring more and more efforts for elaborating an effective theoretical model. In this article, following algebraic mesh operators, we introduce algorithms that perform morphological transformations on unorganized point sets connected by their Delaunay triangulations. We show that these algorithms correspond to morphological operators like erosion, dilation or opening, acting as "shape filters" on meshes. More theoretically, a link is established between these algorithms and the formalisms of edge algebra and α-objects. Then, the mesh operators are defined in terms of complete lattices. These algorithms are applied to the problem -among others -of scene reconstruction by stereoscopy in which objects are represented by unstructured and noisy clouds of 3D points. Rapid prototyping should also benefit from these algorithms.